Harlequins still have plenty to do if they wish to lift the Amlin Challenge Cup this season but they are entering the closing furlongs in decent shape. Wasps launched a spirited fightback on a balmy evening in south-west London but for long periods there was only one possible winner. Quins will now meet the winners of the quarter-final on Saturday between Brive and Munster in the last four.

If the final scoreline looks respectable enough from a Wasps perspective it could have been a very different story. Quins, winners of this tournament in 2001 and 2004, opened up a 14-0 lead inside the first five minutes and briefly threatened to cause mayhem. In the circumstances the injury-hit visitors, who confounded the odds by scoring three tries of their own, deserve credit for salvaging as much pride as they did.

In terms of European qualification next season, however, Wasps are now out of the running, having lost eight of their past nine games in all competitions. It is not the kind of finish the erstwhile Heineken Cup champions are used to but the evidence of collective decline is hard to ignore. It did not help that their captain, John Hart, had to be pulled out of the game after it emerged he was not registered for the tournament.

A predominantly young Quins side, in contrast, is on the up. Beating Munster at Thomond Park may still be an assignment too far but at home, as Leicester found last weekend, they are increasingly doughty opponents. On this occasion it took the hosts 90 seconds to set the tone, Danny Care nipping clear in characteristic fashion to score by the posts.

Wasps had already lost their influential centre Ben Jacobs, who tweaked a hamstring in the warm-up, but there was no excuse for the limp tackling which allowed Maurie Fa'asavalu to score his side's second try and extend the home lead to 14-0 inside five minutes. Nor did Wasps look particularly confident in attack. Even when Richard Haughton seized on turnover ball and raced 40 metres upfield he then chose to kick the ball away in the vain hope someone would chase it. No one did. Humiliation looked a strong possibility.

Maybe that looming prospect was what stirred Wasps into action. Joe Simpson and Riki Flutey combined to put Seb Jewell, Jacobs's replacement, over for a try against his former club and a Dave Walder drop goal pegged the deficit back to four points. The loss of Dave Lemi and Rob Webber before half-time, though, further stretched the visitors' already slender resources and three Nick Evans penalties rewarded Quins' increasing dominance.

Evans, who turned in another casually impressive display, did not miss a single kick but it was Wasps who scored the only try in the third quarter through their replacement hooker Tom Lindsay. If Quins do have a fault it is a tendency not to turn their excellent offloading approach work into points often enough. It cost them against the Tigers and might have done so again here had Wasps started with greater purpose. "We started like a house on fire but we lacked the physicality we normally have in defence," said Conor O'Shea, Quins' director of rugby.

The crowd were certainly twitching entering the final minutes after Haughton had scored wide on the right to drag Wasps back to within a score at 29-22. Flutey, in particular, looked much sharper than he has done, good news for the England management. It required Evans to slot a relieving drop goal and ensure Quins finished the job they had started so ruthlessly. This competition has not always enjoyed centre stage but there is now every prospect of a rousing pair of semi‑finals in three weeks' time. Quins, as the only English representatives left in the draw, would love to go all the way.